The base of this chicken with dumplings is as all-American as they come. But the additional of Parmigiano-Reggiano to the biscuit-y dumplings takes the dish on a trip to the north of Italy.
Season the chicken liberally with salt and pepper. In a large Dutch oven set over medium-high heat, heat the oil until it shimmers. Add half of the chicken thighs, and brown, turning once, about 5 minutes total. Transfer the thighs to a plate and drain the fat. Repeat with the remaining thighs, then drain the fat.
Preheat the oven to 450°. Return the Dutch oven to the stove over medium heat. Add the butter, and when it begins to foam, add the carrots, celery, onion and 1/2 teaspoon salt. Cook, stirring occasionally, until the vegetables soften, about 5 to 7 minutes. Add the wine and stir, scraping the bottom of the pan to remove any stuck bits. Add the flour, broth, milk, thyme, bay leaves and reserved chicken and bring to a simmer. Cover and simmer until the chicken is almost tender, about 45 minutes. Add the peas and continue cooking until the peas are soft and the chicken is completely tender, about 15 minutes. Discard the bay leaves.
Line a baking sheet with parchment paper. In a large bowl, combine the flour, salt and baking soda, then add the Parmigiano-Reggiano and combine. In a small bowl, whisk together the egg, buttermilk and melted butter. Add the buttermilk mixture to the flour mixture and combine with a spatula. Using two spoons or a scoop, scoop golf ball–sized pieces of dough onto the parchment-lined baking sheet, leaving about 1/2-inch space between them. Bake until the tops are golden and the dumplings sound hollow when tapped, about 15 minutes.
Using tongs, transfer the chicken to a cutting board. When it is cool enough to handle, remove the skin and, using two forks, pull the meat from the bones and shred it. Remove much of the fat from the stew by either using a flat spoon or a fat separator. Return the shredded chicken to the pot. When the dumplings are ready, add the parsley to the warm stew and stir. Divide the stew between bowls and top with the dumplings, dunking the dumplings into the stew to soften them. Serve immediately.
Prepare the chicken up to the peas in step 2 and refrigerate up to three days.